In this dissertation we study the algebraic properties of ideals constructed from graphs. We use algebraic techniques to study the PMU Placement Problem from electrical engineering which asks for optimal placement of sensors, called PMUs, in an electrical power system. Motivated by algebraic and geometric considerations, we characterize the trees for which all minimal PMU covers have the same size. Additionally, we investigate the power edge ideal of Moore, Rogers, and Sather-Wagstaff which identifies the PMU covers of a power system like the edge ideal of a graph identifies the vertex covers. We characterize the trees for which the power edge ideal is unmixed, and we show that such ideals are complete intersections. We also characterize the coronas for which the power edge ideal is unmixed, and we show that such ideals are Cohen-Macaulay. For non-trees, we exhibit graphs whose power edge ideals distinguish between the complete intersection, Gorenstein, Cohen-Macaulay, and unmixed properties. We also provide Macaulay2 code that computes the minimal PMU covers and the power edge ideal of a graph
